Area is the length of Ashtekar's triad field

Abstract

The variables introduced by Ashtekar in general relativity have several useful and nice properties. We point out a further feature of these variables: the so(3)-invariant norm of the variable conjugate to Ashtekar’s connection, namely, the inverse densitized triad or the analogue of the electric field, is the area two-form; that is, it is the two-form that gives physical area to any surface. The new variables naturally determine areas in the same way in which the metric tensor naturally determines lengths.